A mixed methods analysis of students' understanding of slope and derivative concepts and students' mathematical dispositions.

摘要

This dissertation examined understanding of slope and derivative concepts and mathematical dispositions of first-semester college calculus students, who are recent high school graduates, transitioning to university mathematics.;The present investigation extends existing research in the following ways. First, based on this investigation, the researcher found that there was a drop in students' mathematical dispositions during week four. From week four to week thirteen there were improved mathematical dispositions; however, the improvement was lower than the baseline disposition scores at week one.;Second, most student responses began calculus at the university with an APOS action level understanding of slope. APOS is a framework for research and curriculum development in undergraduate mathematics education. Students progressed in their understanding of slope throughout the time of the study, however most student responses remained at the action level of understanding slope. For students' understanding of derivative, most student responses began university calculus at either the pre-action or the action level of understanding. Even though students progressed in their understanding of derivative, most student responses remained at the action level.;Third, this investigation found that there was no significant relationship found between changes in students' mathematical dispositions and their knowledge of slope concepts; between students' mathematical dispositions and their knowledge of derivative concepts; or between students' knowledge of slope concepts and their knowledge of derivative concepts.;Fourth, this investigation found that students who participated in the teaching experiment had improved mathematical dispositions at week thirteen compared to week four. Four of the seven students had a higher mathematical disposition at week thirteen than at week one.;Additionally, findings from this study indicate that even though students were able to correctly understand and explain slope and derivative concepts in one context, if the same concept was presented in a different manner, students often were unable to correctly solve the problem situation. In both problem situations the same method or knowledge was necessary. This indicates that students are not connecting concepts from one situation to another.;Lastly, this study found that students, who did not participate in the teaching experiment but had an improved mathematical disposition at week thirteen, attributed their improvement to adjusting to the college environment, increased studying, understanding course expectations, and having a positive disposition.;The dissertation is closed with recommendations for further research into students' understanding of slope and derivative concepts and regarding the role of disposition in students' learning of these concepts. Implications for instruction are included in the conclusions of the dissertation. (Abstract shortened by UMI.).